*The following problem is a generalization of a well known inequality:*

*Problem:*

Let be positive real numbers such that . Prove that

, where is a non-negative real number.

*Solution:*

From Holder’s inequality we get that

, or .

So, it suffices to prove that

.

But the last relation is equivalent to .

Let us denote by the respectively. Then .

So, our inequality takes the form . This inequality is homogeneous so, we consider the sum equal to .

Doing some manipulations in left and right hand side we only need to prove that

.

Now from the AM-GM inequality we have that

.

The last one is equal to .

After that, we only need to prove

,

which is Schur’s 3rd degree inequality, *Q.E.D.*

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